How does transonic flow induce pitching of a NACA0012 airfoil in DNS?

Two regimes split at reduced velocity U*=4

The paper's central comparison is between a low and a high reduced velocity. Reduced velocity U* is defined as U∞/(fN Lc), so a larger U* corresponds to a softer spring and a lower natural pitching frequency. The authors report that for U*≤4 the airfoil behaves almost rigidly with weak coupling to the fluid mode, while for U*>4 the coupled frequency locks onto the natural elastic frequency and the interaction becomes strongly nonlinear.

That split is why the study selects U*=2 and U*=8 as representative cases for the two zones. All simulations use angle of attack α=8°, reduced mass m*=5.0, Reynolds number Re∞=3×10^6, and Mach number M∞=0.8, so the reduced velocity is the variable being changed, not the flow conditions.

The stationary airfoil already oscillates at St=1.316

Before any structural motion is allowed, the fixed airfoil at α=8° produces a dominant non-dimensional frequency of St=1.3161 in the FFT of the moment coefficient. The same dominant Strouhal number, reported as St=1.3160, appears in the lift, drag, and moment spectra, which the authors describe as a high-frequency chaotic flow regime.

The paper attributes this peak to Kelvin–Helmholtz instabilities in the separated shear layer over the airfoil, which roll up into vortices and make the separation bubble breathe at a much higher frequency than the low-frequency buffet motion. The authors also note a low-frequency peak of 0.0841 in the stationary-airfoil spectrum, which becomes important for interpreting the U*=2 case.

At U*=2 the pitching follows a buffet subharmonic

For U*=2, the FFT of the pitching-angle time series gives a coupled frequency of Stc=0.07629. The natural aeroelastic frequency is taken as Stn=1/U*, which is 0.5 at U*=2. The coupled frequency does not sit at the natural value; instead it aligns with the subharmonic of the shock-buffet frequency, and the authors note that the stationary-airfoil low-frequency peak of 0.0841 closely matches the frequency observed at U*=2.

The time series at U*=2 is near-periodic, and the spectrum shows multiple peaks that are not harmonics of the dominant frequency, which the authors call a quasi-periodic response. In this regime the flow mode, not the structure, sets the pitching frequency.

At U*=8 the pitching locks onto the natural frequency

For U*=8, the coupled frequency from the pitching-angle spectrum is Stc=0.1716, while the natural aeroelastic frequency Stn=1/U* is 0.125. The authors state that at high U* the coupled frequency follows the natural frequency of the elastic airfoil, indicating a dominant influence of the structural mode.

The U*=8 time series is highly aperiodic, and its spectrum shows a single dominant frequency with broadened components, which the paper interprets as the onset of chaotic dynamics from strong nonlinear fluid–structure interaction. The authors also report that the high-frequency component seen in the stationary case is absent at both U*=2 and U*=8, suggesting that coupling the shock-buffet mode to the aeroelastic pitch mode diminishes the dominance of shear-layer instabilities.

Peak moment and peak pitch occur at different U*

The maximum moment-coefficient fluctuation |C′M|max peaks at U*=6, but the maximum angular-displacement fluctuation |θ′|max peaks at U*=8. These are different quantities measured on different scales, and the paper explains the offset by the strength of coupling: at U*=6 the pitching frequency approaches the natural buffeting frequency or its harmonics, producing a strong aeroelastic response that modifies the flow and raises |C′M|max, whereas at U*=8 the fluid and structural modes are comparatively weakly coupled and the coupled frequency simply follows the natural frequency, giving the largest |θ′|max.

The lift fluctuation |C′L|max reaches its maximum at U*=7.0, and the drag fluctuation follows a similar trend, although |C′D|max for U*≤4 is higher in the stationary case than in the pitching case because the stationary condition involves a higher effective angle of attack. The mean lift, moment, and angular displacement all increase at low U* as the spring stiffness decreases, and the mean drag rises after U*>4 because it depends dominantly on shock-boundary layer interaction.

Vortex pattern and shock location differ between the two cases

At the instants of peak angular displacement, the U*=2 case shows a 2S vortex shedding pattern, while the U*=8 case shows a P+S pattern. The authors attribute this difference to shock-boundary layer interaction modified by fluid–structure coupling.

The schlieren contours place the shock almost at the trailing edge with a boundary-layer separation zone for U*=2, whereas for U*=8 a λ shock sits at mid chord. Shock-associated noise radiation is much more dominant in the U*=8 case. These visualizations describe the flow at specific time instants (τ=208 for U*=2 and τ=209 for U*=8), not a time-averaged picture.

The 2D DNS bounds how far these results transfer

The study is a 2D direct numerical simulation, and the authors state explicitly that it cannot fully capture inherently 3D buffeting or flutter effects in real airfoils; they plan to extend the simulations to 3D. The results therefore describe a two-dimensional NACA0012 at the single angle of attack, Mach number, Reynolds number, and reduced mass listed above, with reduced velocity varied from 2 to 8.

The solver uses a second-order optimized upwind scheme for spatial derivatives, fourth-order Runge–Kutta time integration, a second-order central difference for viscous terms, and a pressure-based JST switch to capture shocks. The airfoil is pivoted at 0.25c and the far-field boundary sits at 60 chord lengths, choices the authors made to let the airfoil pitch freely and to limit far-field reflections.

The U*=2 and U*=8 cases answer different questions

The most useful way to read this paper is as a two-point comparison rather than a single trend. If the question is whether flow or structure sets the pitching frequency, the answer depends on reduced velocity: at U*=2 the coupled frequency Stc=0.0762 tracks a buffet subharmonic and the motion is flow-dominated, while at U*=8 Stc=0.1716 tracks the natural aeroelastic frequency Stn=0.125 and the structural mode dominates.

If the question is which condition produces the largest structural response, the answer is U*=8 for angular displacement but U*=6 for moment-coefficient fluctuation, so the two peaks should not be treated as the same indicator. The paper's own framing places the transition between the rigid-like and coupled zones at U*=4, which is the threshold a reader should carry forward when comparing these results with other reduced velocities.

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